Two moles of Helium gas (γ = 5/3) are initially at temperature 27 degree Celcius and occupy a volume of 20 litres. The gas is first expanded at constant pressure until the volume is doubled. Then it undergoes an adiabatic change until the temperature returns to its initial value.what is the total change in internal energy of the helium?
To calculate the total change in internal energy of the helium gas, we need to consider two separate processes: the expansion at constant pressure and the adiabatic change.
Let's break down the problem and find the solutions step by step:
Step 1: Expansion at constant pressure
In this step, the gas undergoes an isobaric process, meaning the pressure remains constant. We need to calculate the work done during this process.
Given:
Initial volume, V1 = 20 liters
Final volume, V2 = 2 * V1 = 40 liters (doubled)
For an isobaric process, the work done is given by:
W = P * ΔV
where P is the constant pressure and ΔV is the change in volume (V2 - V1).
Now, we need to find the pressure. According to the ideal gas law:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature (in Kelvin).
We are given:
Number of moles, n = 2
Temperature, T = 27 degrees Celsius = 273 + 27 = 300 Kelvin (converted to Kelvin)
Using these values, we can rearrange the ideal gas law equation to solve for pressure P:
P = nRT / V
Substituting the values:
P = (2 * R * 300) / 20
Step 2: Adiabatic change
In this step, the gas undergoes an adiabatic process, meaning there is no heat exchange with the surroundings. The equation for the adiabatic process is given by:
P1 * V1^γ = P2 * V2^γ,
where P1 and V1 are the initial pressure and volume, P2 and V2 are the final pressure and volume, and γ is the heat capacity ratio for the gas (given as γ = 5/3).
We know the initial and final volumes from the previous step, but we don't know the final pressure. To find it, we can use the ideal gas law as before, but this time we solve for P using the final volume (V2) and temperature (T). The number of moles (n) remains the same.
P2 = (n * R * T) / V2
Step 3: Calculating the change in internal energy
The change in internal energy (ΔU) is given by the sum of the work done (W) and the heat transferred (Q) during the process:
ΔU = Q - W
In an adiabatic process, there is no heat transfer (Q = 0), so the change in internal energy simplifies to:
ΔU = -W
Since we know the work done during the constant pressure process (Step 1), we can substitute this value into the equation.
Finally, calculate the total change in internal energy (ΔUt) by adding the changes from Step 1 and Step 2:
ΔUt = ΔU1 + ΔU2
where ΔU1 is the change in internal energy during the constant pressure process and ΔU2 is the change in internal energy during the adiabatic process.
I hope this explanation helps! Let me know if you need assistance with the calculations.